138 THE EFFECT OF JUNCTIONS ON THE [3h
consider the present and a few analogous problems. Some considerations a more general character are prefixed.
If P, Q, R be components of electromotive intensity, a, b, c those magnetisation, Maxwell's general circuital relations* for the dielectric give
da _ dQ dR /-, \
dt dz dy
and two similar equations, and
.-—
dt \dy dz
also with two similar equations, V being the velocity of propagation. Frc (1) and (2) we may derive
da db dc A dP dQ dR A /<},
_ — (- — 1_ = o — |_ __. _)_____ (j • ............... ^j
dec dy dz dx dy dz
and, further, that (^1_72V2VP, Q, R, a, b, c) = 0, ..... ............... (4)
\Cit /
where ' V2 = d*/dx2 + d^fdy* + d"jdz- ........................ (5)
At any point upon the surface of a conductor, regarded as perfect, t condition to be satisfied is that the vector (P, Q, R) be there normal. what follows we shall have to deal only with simple vibrations in which ; the quantities are proportional to ei:pt} so that d/dt may be replaced by ip.
It may be convenient to commence with some cases where the waves a in two dimensions (x, z) only, supposing that a, c, Q vanish, while b, P, are independent of y. From (1) and (2) we have
.
dx dz
At the surface of a conductor P, Q are proportional to the directi cosines of the normal (n) ; so that the surface condition may be express simply by
l = °. ..................................... 0
which, with . (-*+£ + *.)»_„, .......................... .(,)
suffices to determine b. In (7) k=p/V. It will be seen that equations ( (7) are identical with those which apply in two dimensions to aer vibrations executed in spaces bounded by fixed walls, b then denoti velocity-potential. When. 6 is known, the remaining functions follow once. . . •
* Phil Trans. 1868 ; Maxwell's Scientific Papers, Vol. n. p. 128.ub) tri ,v . .